A radio system includes a transmitter that transmits information-carrying signals to a receiver. The transmitter includes a power amplifier that operates to amplify the signal to be transmitted to a power level that is sufficient to enable receipt of the signal by the receiver. For the power amplifier to achieve high efficiency in terms of the ratio of peak power to average power, the power amplifier of a transmitter is operated in a non-linear region. This causes distortion of the input signal and broadening of the bandwidth of the input signal. To compensate for the distortion of the signal introduced by the power amplifier, the input signal is first passed through a pre-distorter that pre-distorts the input signal.
A typical pre-distorter is itself non-linear, having a non-linearity that compensates for the non-linearity of the pre-distorter. To illustrate, a power amplifier may exhibit first and third order effects characterized by a polynomial function of the input that may be written for third order non-linearities as:y=fNL-IM3(x)=a1x+a3x3  (AW-01)where x is the input signal and a3 is much less than a1. The function f is the response of the power amplifier to the input x and the subscript NL-IM3 denotes non-linearity up to order three. To compensate for the distortion introduced by the power amplifier, a pre-distorter may have a response that is a polynomial function of the input:z=fPD-IM3(x)=b1x+b3x3  (AW-02)
Substituting equation AW-02 into equation AW-01 leads to:y=fNL-IM3(fPD-IM3(x))−a1b1x+(a1b3+a3b13)x3+O(x5)  (AW-03)where O(x5) are terms of 5th order or higher. By appropriate selection of the coefficients b1 and b3, the third order term may be removed at the expense of creating higher order terms of less significant magnitude. The solution to achieve this is given by:b3=−a3b13/a1  (AW-04)Without loss of generality, assume that a1=b1=1. Then the solution to compensate for third order distortions is:b3=−a3  (AW-05)This simple illustration is for third order non-linearities. For higher order non-linearities, the same approach may be taken to cancel the higher order terms. Thus, the pre-distorter is a non-linear device that compensates for the distortion caused by the power amplifier.
The bandwidth of the pre-distorter must be wider than the bandwidth of the input signal depending on the order of inter-modulation to be compensated by the pre-distorter. For example, for third order inter-modulations, the pre-distorted signal occupies about three times the bandwidth of the input signal to the pre-distorter. For fifth order inter-modulations, the pre-distorted signal occupies about 5 times the bandwidth of the input signal. Higher bandwidth implies that the sampling rate of the pre-distorted signal must be higher than the sampling rate of the sampled baseband signal from a modulator to avoid aliasing.
The problem of requiring a high sampling rate due to pre-distortion is exacerbated when the input signal is a dual band signal. Dual band signals are used when multiple wireless communication standards specify transmission using more than one frequency band, or when a single wireless communication standard specifies transmission using more than one frequency band. An up-converted dual band signal has a first continuous band at a first carrier frequency and a second continuous band at a second carrier frequency. The spacing between the carrier frequencies is such that the ratio of the carrier frequency spacing to the maximum individual bandwidth of a first or second band is very high so that a very large sampling rate is needed to avoid aliasing. A very high sampling rate is undesirable since a high clock rate may not be available within the system, and/or is more costly to implement, consumes additional power, etc.
What is needed is a method and system for pre-distorting a dual band signal that does not depend upon a sampling rate that is much higher than the sampling rate of a baseband signal of one of the two bands. More particularly, what is needed is a method and system for determining basis functions for pre-distortion of each band of the dual band signal separately while not requiring a high sampling rate.